The purpose of this note is to give a proof of a theorem of Serre,
which states
that if G is a p-group which is not elementary abelian,
then there exist an integer m and non-zero elements
x1, … xm∈H1
(G, Z/p) such that
formula here
with β the Bockstein homomorphism. Denote by mG
the smallest integer m satisfying the above property. The theorem
was originally proved by Serre [5],
without any bound on mG. Later, in
[2], Kroll showed that
mG[les ]pk−1,
with
k=dimZ/pH1
(G, Z/p). Serre, in [6],
also showed that
mG[les ](pk−1)/
(p−1). In [3], using the Evens norm map,
Okuyama and
Sasaki gave a proof with a slight improvement on Serre's bound;
it follows from their proof (see, for example, [1,
Theorem 4.7.3]) that mG[les ](p+1)
pk−2. However, mG
can be sharpened further, as we see below.
For convenience, write
H*ast;(G, Z/p)=H*(G).
For every xi∈H1(G),
set
formula here